This glossary collects the key concepts, people, and ideas you will meet in this lesson. Use it as a reference while you work through the material, or as a review before assessments. Type in the search box to filter entries.

Introduction and Overview

An earlier lesson gave us measures of disease frequency: prevalence, incidence, risk, and rate. An earlier lesson took the same probabilistic vocabulary and applied it at the level of a single test. This lesson brings these strands together: it combines the disease-frequency vocabulary with the 2×2 contingency logic to produce measures of association, the quantitative comparison between exposed and unexposed groups that is the central output of analytic epidemiology. The four content sections build up from the three ratio measures (this section: risk ratio, rate ratio, odds ratio), through difference measures and exposed-group attributable fractions (a later section), to population-level attributable measures and how each measure relates to study design (a later section), and finally to the hypothesis-testing and confidence-interval machinery that turns each of these point estimates into a defensible inference (a later section).

The section begins with the reason for measuring association, sets out the 2×2 tables from which the ratio measures are computed, defines the risk ratio, the incidence rate ratio and the odds ratio, and closes with the relationships among the three.

Why Measure Association?

Measures of association assess the magnitude of the relationship between an exposure (a potential cause) and a disease. Unlike measures of statistical significance, which are heavily dependent on sample size, measures of association indicate the strength of the effect, that is, how much more (or less) likely disease is in exposed compared to non-exposed groups (Tripepi et al., 2007).

Box 6.1 sets out the difference between the strength of an association and its statistical significance, a distinction to which Section 4 returns when it introduces hypothesis tests and confidence intervals.

A measure of association compares the frequency of disease in exposed and non-exposed groups, so the data must first be arranged in a form that makes this comparison possible. The next part sets out that arrangement.

Data Layout

Depending on study design, disease frequency can be expressed as incidence risk, incidence rate, prevalence, or odds. The formulae later in this lesson are written in terms of the cell labels of Table 6.1, for risk data, and Table 6.2, for rate data. For risk data, the standard 2×2 table is:

Table 6.1. Layout of the 2×2 table for incidence risk data, with exposure status in the columns and disease status in the rows.

For rate data, the denominator is person-time at risk rather than the number of individuals:

Table 6.2. Layout of the table for incidence rate data, giving the number of cases and the person-time at risk in each exposure group.

A quick word on odds, since the third measure below is built from them. The risk of an outcome is the number of cases divided by everyone at risk, meaning cases plus non-cases. The odds of the same outcome is the number of cases divided by the non-cases alone. If 20 of 100 people get sick, the risk is 20/100 = 0.20, but the odds are 20/80 = 0.25. Risk and odds stay close while an outcome is uncommon and pull apart as it becomes common, which is the same fact that lets the odds ratio stand in for the risk ratio only when disease is rare.

With the two layouts in place and the difference between risk and odds defined, the next part introduces the three ratio measures that are computed from these tables.

Three Ratio Measures of Association

The risk ratio, the incidence rate ratio and the odds ratio each divide a measure of disease frequency in the exposed group by the same measure in the non-exposed group. Box 6.2 recalls how HSCI 230 defined the three measures and poses a short retrieval question on them.

The three measures differ in the data they require, and so in the study designs that can supply them. The flip cards below give the formula for each measure and the designs from which it can be computed.

Click each card for what each measure requires of the study design:

Two worked examples show the measures in use. Worked Example 6.1 applies the risk ratio and the odds ratio to a study from Brazil that compared diarrhea among those with and without a water cistern, using the risk layout of Table 6.1.

Worked Example 6.2 turns to rate data in the layout of Table 6.2 and computes the incidence rate ratio for migraine in females and males aged 30 to 40.

The Three Formulae, with Calculators

Each formula below is paired with a calculator that starts from the worked examples above. Entering the cistern data into both the risk ratio and the odds ratio calculators shows how far the two measures separate when the outcome is common.

Equation 6.1 defines the risk ratio, and Calculator 6.1 computes it, starting from the cistern data of Worked Example 6.1. Equation 6.2 defines the odds ratio, which Calculator 6.2 computes from the same data, and Equation 6.3 defines the incidence rate ratio, which Calculator 6.3 computes from the migraine data of Worked Example 6.2.

The risk ratio and the odds ratio give different values for the same data, as the cistern example shows with an RR of 0.58 and an OR of 0.53. The next part explains how the three measures are related and when one of them can stand in for another.

Relationships Among RR, IR, and OR

In general, IR values are further from the null (1) than RR values, and OR values are even further away (Cornfield, 1951; Knol et al., 2008). This can be visualised on a number line, as in Figure 6.1:

Under some conditions the measures come close to one another, and the accordion below explains when the OR approximates the RR, when the RR approximates the IR, and when the OR from a case-control study estimates the IR.

Interactive 6.1 shows the rare-disease condition at work: it computes the RR and the OR from one 2×2 table and plots how far the OR moves from the RR as the outcome becomes more common, so that the effect of outcome prevalence on the gap between the two measures can be seen directly.

This section has explained why association is measured, set out the layouts for risk and rate data, defined the risk ratio, the incidence rate ratio and the odds ratio, and shown that the OR lies furthest from the null and approximates the RR only when the outcome is rare. The Key Takeaways below summarise these points, the knowledge check tests them, and Section 2 turns from ratio measures to difference measures.

Introduction and Overview

An earlier section covered the three ratio measures (RR, IR, OR), that is, how many times more likely disease is in the exposed group compared to the unexposed. This section turns to the parallel set of difference measures, which answer a different question: not how many times more, but how many extra cases occur because of the exposure. Difference measures lead naturally to attributable fractions in the exposed, which quantify how much disease in the exposed group can be attributed to the exposure itself.

The section first contrasts ratio and difference measures, then defines the risk difference and the incidence rate difference, and then turns to the attributable fraction in the exposed and its use as a measure of vaccine efficacy.

Ratio vs. Difference Measures

The ratio measures from an earlier section (RR, IR, OR) tell us the relative strength of association, but they do not indicate the absolute number of cases attributable to the exposure. Difference (absolute effect) measures address this gap by computing how many additional cases occur because of the exposure; the choice between ratio and difference measures is itself a substantive scientific decision rather than a statistical convenience (Greenland & Pearce, 2015; Tripepi et al., 2007).

It helps to name the two scales this contrast lives on. Ratio measures work on a multiplicative scale: a risk ratio of 2 says the exposed risk is the baseline risk multiplied by two. Difference measures work on an additive scale: a risk difference of 0.04 says the exposed risk is the baseline plus four percentage points. The same association can look striking on one scale and slight on the other, so the choice of scale is a real part of the scientific question rather than a matter of presentation.

Box 6.3 explains why both kinds of measure are needed when the public health impact of an exposure is assessed.

The next part defines the first of the difference measures, the risk difference, together with its counterpart for rate data, the incidence rate difference.

Risk Difference (RD): Attributable Risk

The risk difference (RD), also called attributable risk (Walter, 1976), is simply the risk in the exposed group minus the risk in the non-exposed group:

Similarly, the incidence rate difference (ID) is the difference between two incidence rates:

The three tabs below interpret and apply these two formulae. The first tab explains how the sign of a difference measure is read. In the second tab, Worked Example 6.3 applies Equation 6.4 to a cohort of 5,000 women followed through pregnancy, with smoking as the exposure and low birth weight as the outcome. In the third tab, Worked Example 6.4 applies Equation 6.5 to the migraine data of Worked Example 6.2. Calculator 6.4 and Calculator 6.5, which follow the tabs, reproduce the two worked examples so that the effect of changing the counts on each difference can be seen.

Worked Example 6.5 takes the reciprocal of the risk difference from Worked Example 6.3 to obtain the number needed to harm, and applies the same calculation to a protective intervention to obtain the number needed to treat.

The risk difference and its reciprocal describe the excess risk in absolute terms. The next part expresses the same excess as a proportion of the risk in the exposed group, which gives the attributable fraction in the exposed.

Attributable Fraction in the Exposed (AF_e)

The AF_e (also called the attributable fraction among the exposed) expresses the proportion of disease in exposed individuals that is due to the exposure, assuming the relationship is causal (Walter, 1976). It can be viewed as the proportion of disease in the exposed group that would be avoided if the exposure were removed.

Equation 6.6 gives the AF_e in three forms: from the risk difference and the risk in the exposed, from the risk ratio, and approximately from the odds ratio.

AF_e ranges from 0 (where risk is equal, RR = 1) to 1 (where all disease in the exposed group is due to the exposure, RR = ∞). In case-control studies, AF_e can be approximated by substituting OR for RR.

Worked Example 6.6 applies Equation 6.6 to the smoking cohort of Worked Example 6.3, and Calculator 6.6 reproduces the calculation from the two risks, from the RR or from the OR.

Vaccine Efficacy

Vaccine efficacy is a special form of AF_e, where “not vaccinated” is the exposure (factor positive) and “vaccinated” is the comparison group. Worked Example 6.7 works through a simple case, Equation 6.7 then states the general formula, and Calculator 6.7 reproduces the example. If 20% of unvaccinated individuals develop disease versus 5% of vaccinated individuals:

Written as a formula, with R_u the risk in the unvaccinated group and R_v the risk in the vaccinated group:

Box 6.4 distinguishes the AF_e, which measures the excess fraction of cases in the exposed group, from the etiologic fraction, for which the AF_e provides a lower bound.

This section has defined the risk difference and the incidence rate difference, the numbers needed to harm and to treat, and the attributable fraction in the exposed, with vaccine efficacy as a special case. The reflection that follows asks for these measures to be computed for a single exposure, and the Key Takeaways and knowledge check consolidate the section. Section 3 moves from the exposed group to the whole population.

Introduction and Overview

Earlier sections worked at the level of the exposed group. This section zooms out: even if an exposure powerfully causes disease in the exposed, its public-health importance also depends on how common it is in the population. The population attributable fraction (AF_p) captures this combination, and the section closes by mapping each measure of association onto the study design that produces it, tying back to earlier lessons.

The section first defines the population attributable risk and the population attributable fraction, then shows how the fraction is estimated when confounding calls for an adjusted risk ratio, and ends with a summary of the measures that each study design can supply.

From the Exposed Group to the Entire Population

While RD and AF_e describe the effect of exposure among exposed individuals, public health decisions often require understanding the impact of an exposure on the entire population. Two key population-level measures address this:

Population Attributable Risk (PAR)

The PAR is the increase in overall population risk attributable to the exposure. It reflects both the strength of the association and the frequency of the exposure in the population, an idea originally developed by Levin (1953) for lung cancer and reviewed by Northridge (1995) as a link between causal inference and public-health action.

Equation 6.8 gives the PAR in three equivalent forms, the last of which shows that it equals the risk difference multiplied by the proportion of the population that is exposed.

Population Attributable Fraction (AF_p)

The population attributable fraction is the proportion of all disease in the population that is attributable to the exposure. Box 6.5 recalls how HSCI 341 Lesson 1 introduced it, and Equation 6.9 then sets out two ways of computing it.

This section adds a second route to the same quantity: AF_p is the population attributable risk divided by the total population risk, which links it to Equation 6.8.

The second form of Equation 6.9 contains both the exposure prevalence p(E+) and the risk ratio, and Box 6.6 explains why the prevalence of exposure shapes the population impact of a risk factor.

Worked Example 6.8 returns to the smoking and low birth weight cohort of Worked Example 6.3, computes the PAR with Equation 6.8, and divides it by the population risk, as in the first form of Equation 6.9. Calculator 6.8 and Calculator 6.9, which follow the example, reproduce the two results, and Calculator 6.9 also offers presets that compare a common, weak exposure with a rare, strong one.

Worked Example 6.8 used the unadjusted risks of a single cohort, and the AF_p it gives assumes that the association is not confounded. The next part gives the form of the AF_p that is used with a confounding-adjusted risk ratio.

Confounding and AF_p

If confounding is present, adjusted estimates of RR should be used. The AF_p can then be estimated using:

where pd is the proportion of cases exposed to the risk factor, and aRR is the adjusted risk ratio. For multiple exposure categories, a summation formula is used.

Worked Example 6.9 applies Equation 6.10 to a hypothetical cohort study of physical inactivity and type 2 diabetes, and Calculator 6.10 reproduces the example.

The measures in this section and the two before it differ in the data they require. The next part sets out which of them each study design can supply.

Study Design and Measures of Association

Not all measures can be computed from all study designs. Table 6.3 summarises which measures are available:

Table 6.3. Measures of association that can be computed from cross-sectional, cohort and case-control studies.

^a Requires independent estimate of p(D+) or p(E+). ^b Estimated using OR. ^c Requires OR and independent estimate of p(E+|D+).

This section has extended the measures of Section 2 from the exposed group to the whole population, shown how a confounding-adjusted risk ratio enters the AF_p, and matched each measure to the designs that can supply it. The reflection that follows applies Equation 6.9 to two risk factors that differ in strength and prevalence, and the Key Takeaways and knowledge check consolidate the section. Section 4 turns to the precision of these estimates.

Section 4 of 4

Hypothesis Testing & Confidence Intervals

Standard errors, P-values, confidence intervals, and a guide to choosing the right statistical test.

Standard error

Precision of a point estimate

Difference measures (RD, ID)

Variance computed directly from cell counts.

Var(RD)

\[ \frac{\color{#C2410C}{R_{E+}}(1-\color{#C2410C}{R_{E+}})}{\color{#C2410C}{n_1}} + \frac{\color{#1D4ED8}{R_{E-}}(1-\color{#1D4ED8}{R_{E-}})}{\color{#1D4ED8}{n_0}} \]

R_E+, n₁ risk and size, exposed groupR_E−, n₀ risk and size, unexposed group

Ratio measures (RR, IR, OR)

Variance on the log scale using Taylor series approximations.

Var(ln OR)

\[ \frac{1}{\color{#C2410C}{a_1}} + \frac{1}{\color{#BE185D}{b_1}} + \frac{1}{\color{#1D4ED8}{a_0}} + \frac{1}{\color{#6D28D9}{b_0}} \]

a₁ exposed casesb₁ exposed non-casesa₀ unexposed casesb₀ unexposed non-cases

Confidence intervals

Uncertainty around the point estimate

CI for difference measures (symmetric)

\[ \color{#0B7B6B}{\hat{\theta}} \pm \color{#1D4ED8}{z_{\alpha/2}} \cdot \color{#C2410C}{SE(\hat{\theta})} \]

θ̂ point estimatez_α/2 critical valueSE standard error

CI for ratio measures (log scale, then exponentiate)

\[ \exp\!\left[\color{#0B7B6B}{\ln\hat{\theta}} \pm \color{#1D4ED8}{z_{\alpha/2}} \cdot \color{#C2410C}{SE(\ln\hat{\theta})}\right] \]

ln θ̂ estimate on the log scalez_α/2 critical valueSE standard error of the log estimate

Significance criterion: the CI for OR, RR, or IR must exclude 1; the CI for RD or ID must exclude 0.

From the lesson's examples

Computed confidence intervals

Four test statistics

Tests for 2x2 tables and regression

Pearson chi-squared

For categorical outcomes with expected cell counts all above 5. Compares observed to expected counts under independence.

Fisher's exact

Small expected counts (below 5). Exact P-value, no large-sample assumption. Often reported with the OR and its exact CI.

Wald test

Estimate divided by its standard error. Fast, but less accurate in small samples.

Likelihood ratio test

Generally preferred in regression settings; more accurate than Wald in small samples.

Choosing the right test

A decision framework

Carry forward

Point estimates need uncertainty bands

• Standard error quantifies precision; computed differently for difference vs. ratio measures.
• Confidence intervals show the range of plausible effects, not just a binary signal.
• P-values are not the only verdict: a P-value of 0.049 and one of 0.051 carry almost identical information.

The final assessment is just below. Take the synthesis materials slowly before attempting the fifteen questions.

Introduction and Overview

Earlier sections produced point estimates of association: single numbers like RR = 2.5 or AF_p = 30%. Those numbers are useless without a quantification of how uncertain they are. This section closes the lesson by introducing the standard error, hypothesis tests, and confidence intervals, the same statistical-inference machinery you previewed in an earlier course, now applied directly to the measures you just learned to compute.

The section first defines the standard error of each measure, then uses it in hypothesis tests and confidence intervals, and ends with a guide to choosing a statistical test for other kinds of outcome and comparison.

Standard Error

The standard error (SE) provides a measure of the precision of a point estimate, that is, how much uncertainty exists in the estimate. This part gives the variance of the risk difference (Equation 6.11) and of the log risk ratio and the log odds ratio (Equation 6.12 and Equation 6.13), each with a worked example on the smoking and low birth weight cohort and a calculator. For difference measures (RD, ID), the variance can be computed directly:

Worked Example 6.10 applies Equation 6.11 to the smoking cohort, and Calculator 6.11 reproduces the calculation so that the contributions of the two groups to the variance can be compared.

For ratio measures (RR, IR, OR), the variance is computed on the log scale using Taylor series approximations:

The first tab gives the variance of ln(RR) (Equation 6.12), which Worked Example 6.11 applies to the smoking cohort and Calculator 6.12 reproduces. The second tab gives the variance of ln(OR) (Equation 6.13), which Worked Example 6.12 applies to the same cohort and Calculator 6.13 reproduces.

The standard errors computed in this part are used in the Wald statistic and in the confidence intervals later in this section. The next part turns to hypothesis testing.

Hypothesis Testing

A hypothesis test asks how likely the observed data would be if there were no association between exposure and disease. Box 6.7 recalls how HSCI 230 defined the p-value and separated statistical significance from practical significance, and the paragraphs that follow it state the null hypotheses for the measures in this lesson.

For a measure of association, the null hypothesis states that there is no association:

• For difference measures (RD, ID): H₀: θ = 0
• For ratio measures (RR, IR, OR): H₀: θ = 1

An alternative hypothesis can be 1-tailed or 2-tailed. In general, 2-tailed hypotheses are preferred because 1-tailed hypotheses are harder to justify.

Box 6.8 describes what is lost when a P-value is reduced to a verdict of significant or non-significant.

Test Statistics

Four test statistics are in common use for measures of association: the Pearson χ², exact tests, the Wald statistic and the likelihood ratio test. The flip cards below describe each statistic and the conditions under which it applies.

Click each card to explore:

The Pearson χ² and the Wald statistic can both be calculated by hand. The tabs below give each formula with a worked example and a calculator.

The first tab gives the Pearson χ² statistic (Equation 6.14). Worked Example 6.13 applies it to the smoking and low birth weight table, and Calculator 6.14 reproduces the example. The second tab gives the Wald statistic (Equation 6.15), which Worked Example 6.14 applies to the risk difference and the risk ratio of the same cohort, using the standard errors from Worked Example 6.10 and Worked Example 6.11, and which Calculator 6.15 reproduces.

Box 6.8 noted that a confidence interval conveys both significance and precision. The next part defines the confidence interval and shows how it is computed for the difference and ratio measures of this lesson.

Confidence Intervals

A confidence interval (CI) reflects the level of uncertainty in a point estimate. A 95% CI means that if the study were repeated many times under identical conditions, 95% of the computed CIs would contain the true parameter value. This is a property of the procedure, not a probability statement about the parameter (Greenland et al., 2016).

Computing CIs

The method of computing a confidence interval depends on whether the measure is a difference or a ratio. Equation 6.16 gives the interval for a difference measure and Equation 6.17 the interval for a ratio measure, and each is followed by a worked example on the smoking cohort and a calculator.

For difference measures, the CI is computed directly:

Worked Example 6.15 applies Equation 6.16 to the risk difference, and Calculator 6.16 reproduces it.

For ratio measures, the CI is computed on the log scale and then exponentiated:

The CI is symmetrical about lnθ but not about θ itself, which is why confidence intervals for ratio measures appear asymmetric.

Worked Example 6.16 applies Equation 6.17 to the risk ratio and compares the result with the interval for the odds ratio, and Calculator 6.17 reproduces both intervals.

Interpreting CIs

A computed interval can be read in two ways: as a test of the null value and as a range of plausible effect sizes. Box 6.9 sets out both readings for ratio and difference measures, and Box 6.10 applies the first of them to the intervals for the smoking and migraine examples, which Table 6.4 lists.

The next part turns from the measures of association in this lesson to the choice of a statistical test for other kinds of outcome and comparison.

Choosing the Right Statistical Test

The Pearson χ², Fisher’s exact, Wald, and likelihood ratio tests above are the workhorses for 2×2 tables and the regression-based measures of association you will meet in a later course. But epidemiological analyses often require comparing means, proportions, or whole distributions across groups, sometimes paired, sometimes not, sometimes badly skewed. The right test depends on three structural questions about your data:

• Outcome type: continuous (means / medians) or categorical (counts / proportions)?
• Group structure: one group, two groups, or three or more? Independent or paired/matched?
• Distributional assumptions: can you defend approximate normality (or appeal to a large-sample CLT argument), or do you need a non-parametric (rank-based) alternative?

The matrix in Table 6.5 answers those three questions together. Find the row whose outcome type and comparison structure match your data, then read across to the test. The parametric column assumes approximate normality (or a large enough sample to appeal to the central limit theorem), and the rank-based column is the fallback when that assumption cannot be defended, when the sample is small, or when the outcome is ordinal and the spacing between its categories is not meaningful.

Table 6.5. Choosing a statistical test by outcome type, comparison structure and distributional assumptions.

Box 6.11 lists three judgements that remain with the analyst after Table 6.5 has been consulted.

Comparison Table: When Used, How Calculated, How Interpreted

Table 6.6 summarises the tests most commonly reported alongside measures of association.

Table 6.6. When each test is used, how it is calculated and how it is interpreted.

Table 6.6 recommends pairwise comparisons after a significant one-way ANOVA or Kruskal–Wallis test, and Box 6.12 recalls how the significance threshold is adjusted for such multiple comparisons.

Box 6.13 closes the part with a rule that applies to every test in Table 6.5 and Table 6.6: each test is reported together with an effect estimate and its confidence interval.

This section has defined the standard error of each measure, set out the test statistics in common use for measures of association, used the standard error in the Wald statistic and in confidence intervals, and shown how a test is chosen for other kinds of outcome and comparison. The reflection that follows compares two studies by strength of association, statistical significance and precision, and the Key Takeaways and knowledge check consolidate the section.

Bringing It All Together

This lesson translated the disease-frequency measures of an earlier lesson into measures of association between an exposure and an outcome. You worked through the three ratio measures (RR, IR, OR), the difference measures of effect in the exposed (RD, AF_e), the population-level extensions (PAR, AF_p), and the inferential machinery (standard errors, hypothesis tests, confidence intervals) that surrounds every estimate.

Lesson 7, Designing Against Bias, uses these measures to examine validity and confounding in study protocols, and Lesson 8, Time-to-Event Data, applies them to survival curves and hazard ratios. The measures you computed here will reappear in each of those lessons as the outputs that designs deliver and that systematic reviews eventually pool. For deeper treatment of these measures and their statistical foundations, see Greenland and Pearce (2015) and the historical landmark papers by Cornfield (1951) and Doll and Hill (1950).

Final Knowledge Assessment

Complete all 15 questions below with 100% accuracy to finish this lesson. You must also complete the reflection above before submitting.